Transition states and the critical parameters of central potentials

Transition states or quantum states of zero energy appear at the boundary between the discrete part of the spectrum of negative energies and the continuum part of positive energy states. As such, transition states can be regarded as a limiting case of a bound state with vanishing binding energy, emerging for a particular set of critical potential parameters. In this work, we study the properties of these critical parameters for short-range central potentials. To this end, we develop two exact methods and also utilize the first- and second-order WKB approximations. Using these methods, we have calculated the critical parameters for several widely used central potentials. The general analytic expressions for the asymptotic representations of the critical parameters were derived for cases where either the orbital quantum number l or the number n of bound states approaches infinity. The above mathematical models enable us to answer the following physical (quantum mechanical) questions. (i) What is the number of bound states for a given central potential and given orbital quantum number l? (ii) What is the maximum value of l which can provide a bound state for the given central potential? (iii) What is the order of energy levels for the given form of the central potential? It is revealed that the ordering of energy levels depends on the potential singularity at the origin.